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How is the Poisson Distribution derived?

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Original post by Vesniep
Wait is Umbridge from Harry Potter? It's been a long time since watching these movies and don't remember her attitude towards education.
Why are you mean with proofs? They are really important for mathematics . I don't say that this particular proof will help you with your Alevel exams,but if you want to be a mathematician you'd better be more curious ; discouraging him from finding a proof doesn't make sense to me.


You're right, I was just messing. I can't **** with your attitude.
Original post by anoymous1111
I won't need to know it in 2 years time or I can relearn it lal


Posted from TSR Mobile


Cool, but we don't have enough time to relearn everything though
Original post by Callum Scott
Or any other subject for that matter. Fooks me right off. How are you supposed to learn anything if you don't bloody understand it!?


That's true! But the bitter truth is that you might not end up getting good grades for your exams because:

1. The things you learn will not be assessed in your exams
2. The time taken to learn those things will not leave you enough time to learn exam techniques
Original post by Mehrdad jafari
That's true! But the bitter truth is that you might not end up getting good grades for your exams because:

1. The things you learn will not be assessed in your exams
2. The time taken to learn those things will not leave you enough time to learn exam techniques


What I would do for an exam board/education system that actually cared about your understanding of the stuff rather than your knowledge
Original post by Callum Scott
I spent about 4/5 hours learning about combinatorics so I could understand the formula for the Binomial distribution and now I completely understand it, but the next day, we were taught about the Poisson distribution and I have no idea how on Earth someone came up with it.


I'm a bit late to the party here, but perhaps the following might help a bit. I'll try to describe "where the Poisson distribution comes from", from a modern point of view.

The Poisson distribution is a very small tip of a very large iceberg; and the name of that iceberg is called the theory of "stochastic processes". I won't go into detail about what a stochastic process is, as I assume you're looking for intuition here, rather than rigour (correct me if I'm wrong). Roughly speaking a stochastic process is a collection of random things that is indexed by some index set (often taken to be time, modelling the evolution of some process over time). So, two examples:

(i) Consider events that happen "at random" over time, like the blips of a Geiger counter measuring the decay of a radioactive source or like stock market crashes. If you impose a number of conditions (the occurrence of events is independent of each other, the number of occurrences over a period of time is on average proportional to the length of that period and a couple of others I won't go into here), then you get a stochastic process called the "(homogeneous) Poisson process".

(ii) Consider points distributed in some finite area of space (such as the location of trees in a patch of countryside). If you make certain assumptions about how these points are distributed, you get a "spatial Poisson process" which is indexed by the two coordinates that measure where the points are.

(iii) Consider the motion of pollen particles - "Brownian motion" - disturbed by the atomic motions of their surrounding medium. The position of a pollen particle can be described by the so-called "Wiener process", which has the same intimate connection to the normal distribution as the Poisson process has to the Poisson distribution.

The point is that in each of these examples of a stochastic process, certain assumptions are made. These assumptions are often very reasonable from the point of view of what you are trying to model. In the case of the Poisson process, you are looking at random occurrences of events over time that are independent of each other and which occur at a constant rate. The assumptions encode the modelling requirements.

Now here's the crux:

(i) for a Poisson process, it can be shown that (and the link in joostan's post above shows that) the number of events occuring in a fixed interval of time is Poisson distributed with a mean proportional to the length of the time interval.

(ii) The fact that the mean is equal to the variance for such a distribution derived from a Poisson process, is a consequence of the assumptions made for the process.

Following on from (ii), if you observe a process in which measurement suggests that the mean is not equal to the variance, then you can conclude that at least one of the assumptions does not hold. Typically for the sort of things you try to model using Poisson processes, it is the assumption of independence between events. If I take the 2D tree model above (where area takes over from the role of time in the 1D Poisson process) you wouldn't actually expect trees to be randomly distributed in space: their pattern of seeding will probably favour growth near parent trees (an attracting effect); growth near an existing large tree will be stifled (a repelling effect); different patches of ground will have different fertility (an effect that will be reflected in the non-homogeneity of the stochastic process; that is, the mean number of trees per unit area will vary from place to place. etc etc.)

Does that help?

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